Test sequences and formal solutions over hyperbolic groups
Given a group G and a sentence $\sigma$ in the language of groups, we say that $\sigma$ is a true sentence in G (denoted by $G\models \sigma$) if $\sigma$ is true if interpreted in G. The first-order or elementary theory of a group G, denoted by Th(G), is the set of all sentences true in G. The groups whose first-order theory we are mainly interested in, are free and hyperbolic groups. The interest of model theorist in the first-order theory of free groups dates back to the 1940s when Alfred Tarski posed his famous questions, which are now widely known as the Tarski-problems, namely if all non-abelian free groups have the same elementary theory and if this theory is decidable. In 2006 Z. Sela and independently O. Kharlampovich and A. Myasnikov showed that $\Th(F_n)=\Th(F_m)$ for all $m,n\geq 2$, answering Tarski's question to the positive. Subsequently Z. Sela generalized his techniques used in his proof of the Tarski problems to classify all finitely generated groups elementary equivalent to a given torsion-free hyperbolic group. One important step in his analysis of the elementary theory of free and torsion-free hyperbolic groups is a generalization of the following theorem of Merzlyakov.\\ Let $F_k=F(a)$ be a non-abelian free group and suppose that $$F_k\models \forall y\ \exists x\ \Sigma(x,y,a)=1\wedge\Psi(x,y,a)\neq 1.$$ Then there exists a retraction $\pi: \langle x,y,a\ |\ \Sigma(x,y,a)=1\rangle \to F(y)\ast F_k$ and $$F_k\models\exists y\ \Psi(\pi(x),y,a)\neq 1.$$ Sela's generalized Merzlyakov's Theorem covers the case that the universal variables y are restricted to some variety. In our work we prove a version of the Generalized Merzlyakov's Theorem over hyperbolic groups (possibly with torsion). We moreover give an example of a Makanin-Razborov diagram for a group with non-trivial JSJ decomposition and conclude that limit groups are in general not freely subgroup separable. We then classify all JSJ decomposition of doubles of free groups of rank two.
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